The central notion in the semantic or model theoretic exploration of modal logic. A possible world is here considered to be a complete state of affairs, or one in which every proposition under consideration has a definite truth-value. A proposition p is necessary (·p) if it is true at all possible worlds, possible (◊p) if it is true at some, and impossible if it is true at none. Various relations can be defined over possible worlds. For example, to model questions arising over the iteration of modal operators, such as whether ·p implies ··p, Kripke defined an accessibility relation over worlds, enabling him to transform this question into the issue of whether, if a proposition is true at all worlds accessible from this, then it is true at all worlds accessible from all worlds accessible from this. To give a theory of counterfactuals, David Lewis used a similarity relationship, so a counterfactual conditional ‘If p had been true, q would have been’ can be regarded as true if it is true at all of the most similar worlds to ours in which p is true.
Although the utility of possible worlds models in exploring modal logic is beyond doubt, the philosophical propriety of the notion has been intensively debated, with the same kinds of position emerging as in the philosophy of mathematics: Platonism, alleging their reality (see modal realism), forms of constructivism, alleging that we create them, and formalism, regarding the notation as a useful tool for logic but nothing more.